Phase behavior in thin films of cylinder-forming ABA block copolymers:
mesoscale modeling
Horvat, A.; Lyakhova, K.S.; Sevink, G.J.A.; Zvelindovsky, A.V.; Magerle, R.
Citation
Horvat, A., Lyakhova, K. S., Sevink, G. J. A., Zvelindovsky, A. V., & Magerle, R. (2004). Phase
behavior in thin films of cylinder-forming ABA block copolymers: mesoscale modeling. Journal
Of Chemical Physics, 120(2), 1117-1126. doi:10.1063/1.1627325
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A. Horvat, K. S. Lyakhova, G. J. A. Sevink, A. V. Zvelindovsky, and R. Magerle
Citation: The Journal of Chemical Physics 120, 1117 (2004); doi: 10.1063/1.1627325 View online: https://doi.org/10.1063/1.1627325
View Table of Contents: http://aip.scitation.org/toc/jcp/120/2 Published by the American Institute of Physics
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Phase behavior in thin films of cylinder-forming
ABA
block copolymers:
Mesoscale modeling
A. Horvat
Physikalische Chemie II, Universita¨t Bayreuth, D-95440 Bayreuth, Germany
K. S. Lyakhova, G. J. A. Sevink, and A. V. Zvelindovsky
Leiden Institute of Chemistry, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlands
R. Magerlea)
Physikalische Chemie II, Universita¨t Bayreuth, D-95440 Bayreuth, Germany 共Received 24 June 2003; accepted 25 September 2003兲
The phase behavior of cylinder-forming ABA block copolymers in thin films is modeled in detail using dynamic density functional theory and compared with recent experiments on polystyrene-block-polybutadiene-block-polystyrene triblock copolymers. Deviations from the bulk structure, such as wetting layer, perforated lamella, and lamella, are identified as surface reconstructions. Their stability regions are determined by an interplay between surface fields and confinement effects. Our results give evidence for a general mechanism governing the phase behavior in thin films of modulated phases. © 2004 American Institute of Physics.
关DOI: 10.1063/1.1627325兴
I. INTRODUCTION
Block copolymers self-assemble into ordered structures with characteristic lengths determined by the molecular size, which is in the 10–100 nm range.1,2 This property has at-tracted much interest in the area of soft condensed matter physics and nanotechnology. There is large interest to under-stand, predict, and control structure formation in this class of ordered polymeric fluids.
In bulk, the block copolymer microdomain structure is determined mainly by the molecular architecture and the in-teraction between the different components 共blocks兲. At the air–polymer interface and the film–substrate interface addi-tional driving forces for structure formation exist. Typically, one component has a lower interfacial energy than the other. This causes a preferential attraction of one type of block to the interface共or surface兲, which can result either in an align-ment of the bulk structure at the interface3–5and/or a devia-tion of the microdomain structure from the bulk. These de-viations in the vicinity of the interface have been shown to be analogous to surface reconstructions of crystal surfaces.6 In thin films, additional constraints play an important role. Here, the microdomain structure has to adjust to two interfaces and a certain film thickness, which can be a non-integer multiple of the microdomain spacing in the bulk. Both constraints together cause a complex and interesting phase behavior.
Since the seminal work of Anastasiadis et al.,3 the be-havior of lamellae forming block copolymers in thin films has been studied in detail and two major effects have been identified 共for reviews, see Refs. 7–9兲. The preferential at-traction of one type of block to the surface共the surface field兲
causes the lamella to align parallel to interfaces and the film forms islands or holes where the film thickness is a 共half兲 integer multiple of the lamella spacing in the bulk. In cases where the film thickness is not compatible with the natural bulk domain spacing or when the film/air and the film/ substrate interface is not selective, lamellae can orient per-pendicular to the interfaces.4,5
The behavior of cylinder forming systems is more com-plex and less understood. Here, the natural hexagonal pack-ing of cylinders cannot be retained close to the planar inter-face, which, regardless of its orientation, always breaks the symmetry of the bulk structure. As a result, besides cylinders oriented parallel and perpendicular to the surface,10–13a va-riety of deviations from the bulk structure have been ob-served near surfaces and in thin films, such as a disordered phase,10 a wetting layer,14 spherical microdomains,15 a per-forated lamella,15as well as more complicated hybrid struc-tures such as cylinders with necks,16 a perforated lamella with spheres,17and an inverted phase.18
Various theories have been used to describe this behavior.19–26A brief summary of experimental and simula-tion results is given in Ref. 24. With dynamic density func-tional theory 共DDFT兲, a dynamic variant of self-consistent field theory, Huinink et al.21,22 have calculated a phase dia-gram for thin films of cylinder forming diblock copolymers. They found that noncylindrical structures are stabilized by the surface field in the vicinity of surfaces and in thin films. With increasing strength of the surface field a sequence of phase transitions was predicted: from a wetting layer, to cyl-inders oriented perpendicular to the surface, to cylcyl-inders ori-ented parallel to the surface, to a hexagonally perforated lamella, and to a lamella.
Theoretical and experimental results agree qualitatively only in part. From the experimental point of view, only single deviations from the bulk structure and no phase
dia-a兲Author to whom all correspondence should be addressed. Electronic mail:
robert.magerle@uni-bayreuth.de
JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 2 8 JANUARY 2004
1117
grams have been reported. Therefore it remains unclear which of the reported phenomena are specific to the particu-lar block copolymer and/or route of film preparation and which are general behavior. From the modeling point of view, no model predicts all experimentally observed phases. In particular, a detailed and quantitative comparison between modeling and experimental results is missing. The underly-ing fundamentals remain unclear.
Recently, Knoll et al.27 have measured the phase dia-gram for thin films of concentrated solutions of a polystyrene-block-polybutadiene-block-polystyrene 共SBS兲 triblock copolymer in chloroform as function of film thick-ness and polymer concentration. In this communication27we have presented matching computer simulations of thin films of ABA triblock copolymers which model in stunning detail the experimentally observed phase behavior. As an example, Fig. 1 shows a comparison of the experimental results of Knoll et al.27 and our simulations of a corresponding
A3B12A3 triblock copolymer film, where the interfaces
pref-erentially attract the B block. In Fig. 1 the film thickness increases from left to right and a rich variety of structures is observed. With increasing film thickness, both experiments and calculations show the same sequence of thin film phases: a disordered film共dis兲 for the smallest thickness, A spheres or very short upright A cylinders (C⬜), A cylinders oriented parallel to the film plane (C储,1), a perforated A lamella共PL兲, parallel oriented A cylinders with an elongated cross section and necks, perpendicular oriented A cylinders (C⬜), and fi-nally two layers of parallel oriented A cylinders (C储,2). The phase transitions occur at well-defined film thickness as can be seen from the white contour lines that represent points of equal film thickness 关Figs. 1共a兲 and 1共b兲兴.
As measured three-dimensional volume images of a thin film’s microdomain structure are rather rare,16,17,28,29 DDFT simulations as shown in Fig. 1共d兲 facilitate us to interpret the experimentally easy observed surface structures in terms of
the volume structure of the films. Furthermore, compared to the experiments27,30 and previous simulations on diblock copolymers,21,22our simulations cover a much larger param-eter space. Only this enabled us to distinguish between the different physical phenomena governing the phase behavoir in thin films. The phenomena and their interplay can be sum-marized in the following way:共1兲 The surface field can either orient the bulk structure or it can stabilize deviations from bulk structures, such as wetting layer, perforated lamella, and lamella, which we identified as surface reconstructions.6共2兲 The film thickness is modulating the stability regions of the different phases via interference of surface fields and con-finement effects.
The aim of this paper is to give a detailed report of our simulation results. The experimental part is reported in the preceeding companion article.30 First, we report briefly the phase behavior of an A3B12A3 melt in the bulk. Then we
focus on the phase behavior of cylinders forming systems in thin films. The basic types of surface structures and surface reconstructions are introduced and the underlying physics is explained. Finally, we compare our results with the corre-sponding experiments of Knoll et al.30and the phase behav-ior of other cylinder and lamella forming block copolymers.
II. METHOD
We have modeled the phase behavior in thin films with mean-field DDFT, which was developed by Fraaije et al.31–33 for mesoscale modeling the phase separation and ordering processes of multicomponent polymeric systems. For our simulations we used the standardMESODYNcode.34
As a molecular model an ideal Gaussian chain is used. In this ‘‘spring and beads’’ model, springs mimic the stretching behavior of a chain fragment and different kinds of beads correspond to different components in the block copolymer. All nonideal interactions are included via a mean field and the strength of interaction between different components is characterized by the interaction parameter ⑀AB, which we
express in units of kJ/mol. This parameter can be related to the conventional Flory–Huggins parameter 共see Sec. III A兲. Interfaces are treated as hard walls, with the flux per-pendicular to the interface and the density at the interface kept equal to zero.33 The interaction between components and interfaces is characterized by corresponding mean field interaction parameters ⑀A M and⑀B M. As only the difference
between the interaction parameters is relevant for structure formation,33we used an effective interface–polymer interac-tion parameter ⑀M⫽⑀A M⫺⑀B M, which characterizes the
strength of the surface field. A positive⑀IJ parameter
corre-sponds to a repulsion between the components I and J. The dynamics of the component densitiesI(r,t), with I⫽A, B, is described by a set of functional Langevin equations. These are diffusion equations of the component densities which take into account the noise in the system. Driving forces for diffusion are local gradients of chemical potentials I
⫽␦F关兵I其兴/␦I. The Langevin diffusion equations are
solved numerically with homogeneous initial conditions. As MESODYNis based on the same type of free energy functional as self-consistent field theory共SCFT兲,35 it is expected to ap-proach on long time scales the same solutions as SCFT does FIG. 1.共a兲, 共b兲 TappingMode™ scanning force microscopy phase images of
thin polystyrene-block-polybutadiene-block-polystyrene共SBS兲 films on sili-con substrates after annealing in chloroform vapor. The surface is covered with a homogeneous⬇10-nm-thick PB layer. Bright 共dark兲 corresponds to PS 共PB兲 microdomains below this top PB layer. Contour lines calculated from the corresponding height images are superimposed. 共c兲 Schematic height profile of the phase images shown in共a兲 and 共b兲. 共d兲 Simulation of an
A3B12A3block copolymer film in one large simulation box with from left to right increasing film thickness H(x),⑀AB⫽6.5, and⑀M⫽6.0. The isodensity
by searching for the absolute minimum of the free energy. With MESODYN, however, structure formation proceeds via local gradients of chemical potentials that are intrinsic to the system. In this way, long lived transition states can also be visited in a simulation run. This ambiguity, however, is shared with the experiments, where the specimen is also quenched after a finite annealing time.30
The simulations were done on a cubic grid of dimen-sions X⫻Y⫻(H⫹1), with the interface positioned at z⫽0. Due to the periodicity of boundary conditions, the system is confined between two interfaces separated by H grid points. The triblock copolymer is modeled as a melt of A3B12A3
chains, which can be seen as two connected A3B6 chains. The architecture of the A3B12A3 chain enters specifically in the calculation of the density fields from the external poten-tials and in the partition function, respectively, the free energy.31–33For our simulations we partly relied on previous results 共Refs. 21 and 22兲. Apart from the chain architecture all simulation parameters are the same as for the A3B6
diblock system studied in Refs. 21 and 22, with an exception for the interaction parameter ⑀AB. In addition, we have
var-ied this interaction parameter in a range where the A3B12A3
system forms cylinders in the bulk. Doing so we can also study the influence of the molecular architecture on the ob-servered phenomena by comparing our results on A3B12A3
triblock copolymer with the behavior of the corresponding
A3B6 diblock copolymer. At the same time, this study
al-lowed us to determine the value of the interaction parameter ⑀ABthat best matches the experimental situation. As in Refs.
21 and 22, we followed the temporal evolution in the system until significant changes of the free energy, the order param-eter, and the microdomain structure no longer occurred.
III. RESULTS A. Bulk structure
As a first step, we parametrized the system studied ex-perimentally by Knoll et al.27,30For this, we investigated the phase behavior in the bulk and determine the range of ⑀AB
where the system forms cylinders. In Fig. 2 the phase dia-grams of a melt of ABA triblock copolymers with A-volume fraction fA⫽1/3 are shown, which were calculated with
DDFT and SCFT.36The Flory–Huggins parameterand⑀AB
are related through ⫽(⫺1/2kT)关2⑀AB⫺⑀AA⫺⑀BB兴 关Eq.
共32兲 in Ref. 32兴. In our case, with ⫽1, ⑀AA⫽⑀BB⫽0, N
⫽18, and T⫽413 K, N⬇⑀AB⫻5.43 mol/kJ.
The phase separation process was simulated with DDFT in a cubic box with 32⫻32⫻32 grid points and periodic boundary conditions. The calculations were started with a homogeneous melt. During the simulation run we observe similar processes as previously reported.31 First the segrega-tion of the A and B blocks into interconnected domains takes place. The separation process continues with the break-up of an initially connected network of different domains into well-defined structures. Microdomains with different orienta-tions form simultaneously during the phase separation pro-cess, which leads to a very defective structure. The last and slowest process is the long-range ordering of the micro-domain structure, which proceeds via annealing of defects.
Our calculations were done until 4000 time steps, where the long-range ordering process is still not completed. Neverthe-less, the result is sufficient to characterize the formed micro-domain structure. For⑀AB⭐5.75 no phase separation occurs.
The A density A is spatially inhomogeneous with a mean
value 共averaged over all grid points兲 of 0.33 and standard deviation of 0.03. The mean value of 0.33 corresponds to the volume fraction fA of the A component. As the interaction
parameter ⑀AB increases, A-rich domains of spherical shape
共S兲 form in a B-rich matrix for 5.8⭐⑀AB⭐6.0. In the range
6.1⭐⑀AB⭐6.5, well-separated cylindrical A microdomains
共C兲 embedded in the B matrix were observed. For ⑀AB
⭓6.6, we observe an A-rich network of microdomains
em-bedded in a B-rich matrix. Because of the large amount of three-fold connections we relate this structure to a defective gyroid phase. We have also done simulations in smaller simulation boxes, 16⫻16⫻16 in size, and obtained similar results but with better ordered structures.
For⑀AB⫽6.5 we determined the distance between
cylin-ders in the bulk. For this purpose we did a simulation in a 64⫻64⫻1 large box, analogous to Huinink et al.21Here, due to the periodic boundary conditions, the cylinders orient per-pendicular to the 64⫻64 plane and show up as hexagonally packed dots. The distance between next-nearest cylinders was determined to be a0⫽6.9⫾0.5 grid units.
Our results are similar to those obtained with SCFT. With increasing interaction parameter both methods predict transitions from a disordered state to spheres, then cylinders, and a gyroid phase. We observe the transition from the dis-ordered state to spheres at a higher value of ⑀AB than
pre-dicted by Matsen et al.36The discrepancy could be due to the relatively small size of the chain and the nonlocality of the nonideal interactions.32The phase boundary between the cyl-FIG. 2. Bulk phase diagrams for ABA triblock copolymer melts with fA
⫽1/3 as a function of interaction parametersN and⑀AB. Results of
Mat-sen共Ref. 30兲 obtained with SCFT are compared with our results calculated with DDFT for an A3B12A3 melt in 32⫻32⫻32 large simulation boxes. Phases are labeled as G 共gyroid兲, ‘‘G’’ 共gyroid-like兲, C 共cylinders兲, S
共spheres兲, and ‘‘dis’’ 共disordered兲.
inder and gyroid phase is the same in both simulation results and this region of the phase diagram is of particular interest of the present study. An obvious difference is the presence of defects in the microdomain structures simulated with DDFT, which Matsen’s SCFT does not take into account. If defects cost very little energy a rather high density of defects might be thermally excited in the system.
B. Surface reconstructions
We now turn to the question of what happens when in-terfaces are added to the system. On varying the film thick-ness, H, and the strength of the surface field,⑀M, we observe
a complex phase behavior. The presence of interfaces has several effects. One is a speed-up of the long range order formation. In Fig. 3 two systems with different boundary conditions and otherwise identical parameters are compared: an A3B12A3melt with⑀AB⫽6.5 in the bulk 关Fig. 3共a兲兴 and in
a film with H⫽54 关Fig. 3共b兲兴. The surface field was chosen to be⑀M⫽6. In both systems, the simulation time was 4000
time steps and both show cylinders. In the film, the temporal evolution of structure formation is similar to that of a bulk system. In addition, however, the cylinders start to align at the interfaces and the alignment propagates from the surface
into the film. This causes the cylinders in the film to orient parallel to the surface and to pack in a neat hexagonal array
关Fig. 3共b兲兴. In the bulk, however, the microdomain structure
is still very defective 关Fig. 3共a兲兴. Although the simulation box of the film system is larger than that of the bulk simu-lation it shows a higher degree of long-range order.
The most intriguing effect of the presence of interfaces are deviations from the bulk microdomain structure in the vicinity of the interface. This effect is called surface recon-struction and it is best seen at large film thicknesses, for instance at H⫽54 共Fig. 4兲. In such films the interfaces are separated by approximately nine layers of cylinders and in the vicinity of one interface the influence of the other one is negligible. In the middle of the film, in most cases the mi-crodomain structure remains hexagonally ordered cylinders aligned parallel to the film plane. Depending on the strength of the surface field, considerable rearrangements of micro-domains near the interfaces, i.e., surface reconstructions, oc-cur. For⑀M⬍2, the A component is preferentially attracted to
the interface and a wetting layer 共W兲 is formed. When ⑀M
increases, cylinders oriented perpendicular to the surface are stabilized for⑀M⬇3. As⑀M is further increased, the A com-ponent is weakly repelled from the interface and cylinders orient parallel to the surface in the range ⑀M⬇4 – 9. For
larger ⑀M, surface reconstructions with noncylindrical
mi-crodomains are induced: first, at ⑀M⬇10, a transition to a
perforated lamellae 共PL兲 occurs in the layer next to the sur-face which transforms to a lamellae 共L兲 at⑀M⬇25.
For the surface structures shown in Fig. 4 we examined the density distribution of each component. In Fig. 5共a兲 the (x,y ) plane averaged A density
具
A典
x,yis plotted as functionof the distance z. For all three ⑀M values共5, 10, and 30兲, a
modulation is observed, which corresponds to a layered mi-crodomain structure oriented parallel to the interface. In all three of the displayed cases the B component is attracted to the interface. This causes a depletion of the A component at the interface and an increase of A in the middle of the first A microdomain next to the interface. The effect increases
with increasing surface field⑀M and it is accompanied with formation of different surface reconstructions. At z⫽3, ap-FIG. 3. Effect of the surface on the long range ordering process. Simulations
for a cylinder forming system with interaction parameter ⑀AB⫽6.5 after
4000 time steps.共a兲 In the bulk, in a 32⫻32⫻32 large simulation box. 共b兲 In a confined film, where X⫽Y⫽32 and H⫽54, surfaces at z⫽0 and 55, and the effective surface–polymer interaction parameter⑀M⫽6.
FIG. 4. Effect of the strength of the surface field⑀Mon microdomain structures and surface reconstructions. Simulation results for an A3B12A3melt (⑀AB
⫽6.5) in a rather thick film (H⫽54) with surfaces at z⫽0 and 55 at⑀M⫽⫺4, 3, 7, 12, 30. Isodensity surfaces (A⫽0.45) are shown for typical structures.
proximately in the middle of the first A-rich microdomain next to the interface, A increases with increasing surface
field⑀M from A⫽0.55 for cylinders (⑀M⫽5), to A⫽0.62
for the perforated lamella (⑀M⫽10), and 0.70 for the lamella
surface reconstruction (⑀M⫽30). In Fig. 5共b兲, the lateral
dis-tributions of the A density at z⫽3 are plotted as histograms for the same values of the surface field as in Fig. 5共a兲. Re-sults for ⑀M⫽5 show a broad density spectrum with two
peaks, which correspond to the presence of two microphase separated components: A-rich cylinders and the B matrix. For ⑀M⫽12, the distribution is still broad and A is the majority
component in this layer, the isodensity surface is a perforated lamella. For⑀M⫽30, the density distribution shows one nar-row peak, as expected for a lamella. These results indicate that with increasing surface field the density variations par-allel to the interface are suppressed in the vicinity of the interface. In these structures the averaged mean curvature is gradually decreased in order to adopt to the planar symmetry of the interface.
C. One microdomain thick films
We now turn to the effect of the film thickness H. In thinner films two additional factors influence the micro-domain structure in the film: the interference of the two sur-face fields共of the bottom and the top interface兲 and the com-mensurability of the natural domain spacing with the film thickness. First, we present the interference effect of surface fields for H⫽6, which corresponds to one layer of cylinders
共Fig. 6兲. For this thickness we observe similar structures as
in thick films (H⫽54) in the vicinity of the interface. For
H⫽6, however, the strength of surface field needed to form
noncylindrical microdomains is strongly reduced. We
ob-FIG. 5. Effect of the strength of the surface field on the distribution of A density.共a兲 The laterally averaged A density具A典x,yas function of the
dis-tance z from the surface 共depth profiles兲 for different surface fields and surface reconstructions: 共䊊兲 parallel cylinders, ⑀M⫽5; 共䉭兲 perforated
lamellae,⑀M⫽10; 共䊐兲 lamellae,⑀M⫽25. 共b兲 Histograms of the lateral
av-eraged A density具A典x,yat z⫽3, approximately in the middle of the first
A-rich microdomain next to the surface, for the surface fields shown in共a兲.
FIG. 6. Simulation results for a cylinder forming A3B12A3melt (⑀AB⫽6.5) in thin films (H⫽6) with different strength of the surface field⑀M. Isodensity
profiles共⫽0.45兲 for typical structures are shown. Gray boxes indicate parameters were simulations have been done.
serve the lamella phase already for⑀M⫽7, compared to ⑀M
⫽25 for H⫽54. Also the perforated lamella phase appears
already at ⑀M⫽6 instead of ⑀M⫽10 and it has a much
smaller existence range. Perpendicular cylinders, which at
H⫽6 are very short and almost spheres, appear in both cases
at⑀M⫽3. An additional feature of thin films is the presence
of a disordered phase with no well-defined microdomain structure, however, with the two components A and B being
still slightly segregated. Figure 7 shows depth profiles of the laterally averaged A density for different structures in thin films of thickness H⫽6. For⑀M⬍1, the A block is
preferen-tially attracted to the surface and a wetting layer forms at each surface. For 1⭐⑀M⭐2 the disordered phase forms and
the A component is only weakly attracted to the interface. Interfaces with ⑀M⫽3 appear as neutral. For this surface
field value very short cylinders oriented perpendicular to the surface are formed. The fact that the interface appears neutral at⑀M⫽3 and not at⑀M⫽0 can be explained by an entropic
attraction of the shorter A block to the interface.21 For ⑀M⬎3 the surface preferentially attracts the B block and A-rich microdomains form in the middle of the film.
D. Phase diagrams of surface reconstructions
We have done simulations with ⑀AB⫽6.3, 6.5, and 7.1
and have varied the strength of the surface field,⑀M, and the
film thickness, H. We have also calculated a phase diagram where we varied ⑀AB and⑀M simultaneously while keeping
⑀AB⫽⑀M. Figures 8 and 9 show the phase diagrams of
sur-face reconstructions for ⑀AB⫽6.5 and 6.3, respectively. For
both values cylinders are formed in the bulk 共see Fig. 2兲 as well as in the middle of the films. Both phase diagrams clearly show that microdomain structures oriented parallel to the surface are dominant. Cylinders orient perpendicular to the surface for the neutral surfaces at ⑀M⬇3 and at certain thicknesses (H⫽9 and 15兲 which strongly deviate from an integer multiple of a natural layer thickness. For hexagonally packed cylinders the natural thickness is c0⫽a0
冑
3/2, where FIG. 7. Effect of the strength of the surface field on the depth profiles of thelaterally averaged A density具A典x,yin thin films. Depth profiles are shown
for:共䊐兲 a lamella at⑀M⫽9, 共䉭兲 a perforated lamella at⑀M⫽7, 共䊊兲
cylin-ders oriented parallel to the surface at⑀M⫽5, 共䊉兲 cylinders oriented
per-pendicular to the surface at⑀M⫽3, 共夝兲 a disordered phase at⑀M⫽2, and
共䊏兲 a wetting layer at⑀M⫽⫺2.
FIG. 8. Phase diagram of surface reconstructions calculated for an A3B12A3melt with⑀AB⫽6.5 as function of film thickness H and surface field⑀M. Boxes
a0is the distance between next-nearest cylinders in the bulk.
In our case共see Sec. I A兲, a0⫽6.9⫾0.5 and the natural
thick-ness of one layer of cylinders is c0⬇6.
Interference of surface fields. The important feature of
the thin film phase behavior is the existence of surface re-constructions with noncylindrical morphologies: the wetting layer, the perforated lamella, and the lamella. For thick films with H⬎3c0 the critical surface field required to induce a surface reconstruction is independent of the thicknesses. For thinner films, this threshold value decreases: for the perfo-rated lamella,⑀M⬇10, 8, 7, and 6 at H⫽9c0, 3c0, 2c0, and c0, respectively. This indicates that surface fields extend into
the bulk with a decay length of about one microdomain spac-ing. Furthermore, they are additive and for very thin films the effects of both surfaces combine. This explains why in thin films a weaker surface field is sufficient to form a PL共or L兲 than in thick films. It also explains the formation of a PL beneath a wetting layer for⑀M⫽0 at H⫽12 and⑀M⫽⫺2 at H⫽19.
Confinement effects modulate the stability regions of
phase oriented parallel to the interfaces. An integer multiple of a natural layer thickness is energetically favored. This causes easier deformable phases to occur at intermediate film thicknesses. For very small thicknesses (H⬍c0) and weak
surface fields, confinement prevents microphase separation and stabilizes a disordered phase.
The phase diagram for⑀AB⫽6.3 共Fig. 9兲 displays a very
similar behavior to the one for ⑀AB⫽6.5 共Fig. 8兲. The two
main differences between the two phase diagrams is that for ⑀AB⫽6.3 the stability region of the disordered phase is larger
and that the threshold values for the formation of surface reconstruction are shifted to larger strengths of the surface field, in particular for the lamella surface reconstruction.
Order of phase transitions. An important feature of our
simulations is the coexistence of different phases in one layer. Figure 10 shows such a situation where parallel cylin-ders and a perforated lamella coexist. This simulation was done until 11 000 time steps and after 5000 time steps no significant changes were observed. The coexistence of phases corresponds nicely to the experimental observation
关see Fig. 1共b兲 and Refs. 27 and 30兴. The presence of
coex-istent phase clearly indicates a first-order phase transition.
The same is also valid for the PL to L transition. The dashed lines in the phase diagrams denote continuous transitions be-tween the W and ‘‘dis,’’ and the ‘‘dis’’ and L phase.
E. Structured wetting layer
A result not displayed in the phase diagrams is the struc-ture of the wetting layer. For thin films (4⭐H⭐8), where the entire film consists only of two wetting layers, the wet-ting layer has no lateral structure. However, in thicker films and for small values of the surface field (⫺1ⱗ⑀Mⱗ2) the wetting layer has a structure which is complementary to the microdomain structure next to it in the middle of the film
共Fig. 11兲. The entire structure shown in Fig. 11共a兲 is very
similar to hexagonally packed cylinders.37In Fig. 12, histo-grams of the lateral density distributions within the wetting layer are shown for different values of the surface field ⑀M.
For⑀M⫽1 and 2, two peaks appear in the histogram which
correspond to a lateral microphase separation, for example, stripes of A and B density. For smaller values ⑀M, the two
peak merge, which reflects the fact that the structure continu-ously transforms to a homogeneous wetting layer. Its histo-gram is similar to that in the middle of a lamella at ⑀M
⫽25 关see Fig. 5共b兲兴 which supports an interpretation as a half
lamella.
IV. DISCUSSION
A. Mapping to the experimental phase diagram
Our simulations reproduce all essential features of the experimentally observed phase behavior of thin films of polystyrene-block-butadiene-block-polystyrene 共SBS兲 tri-FIG. 9. Same as Fig. 7 but for⑀AB⫽6.3.
FIG. 10. Coexistence of parallel cylinders and perforated lamellae for an
A3B12A3 melt with⑀AB⫽6.5,⑀M⫽6.0, and H⫽7. The isodensity level is A⫽0.45. The size of the simulation box is 64⫻64⫻8.
FIG. 11. Structured wetting layers for an A3B12A3melt,⑀AB⫽6.5
关simula-tion box 32⫻32⫻(H⫹1), after 2000 time steps兴. 共a兲 H⫽12,⑀M⫽2, and
isodensity level A⫽0.45; the wetting layer resembles half-cylinders. 共b兲
H⫽12,⑀M⫽0, and isodensity levelA⫽0.45; the wetting layer consists of
isolated dots.共c兲 A perforated wetting layer at H⫽9,⑀M⫽⫺1, and
isoden-sity levelA⫽0.6.
block copolymers studied by Knoll et al.27,30 In particular, the sequence of phases as function of film thickness is cor-rectly modeled. This is nicely seen in Fig. 1共d兲 where a simu-lation done in a wedge-shaped geometry is shown. Also the phase diagrams shown in Figs. 8 and 9 nicely match the experimental one共see Fig. 3 in Ref. 27兲, indicating that the experimental control parameter, the polymer concentration
⌽P, is directly related to the control parameter in the
simu-lations, namely the surface field⑀M.
In order to keep the model as simple as possible we chose to model the SBS/chloroform solution as a melt of
A3B12A3 block copolymer. As chloroform is a nonselective solvent it acts as a plasticizer, which merely induces chain mobility.27,30 The nonselective solvent chloroform screens the interaction between the block copolymer components and the interfaces. This effect is modeled by interaction param-eters ⑀AB and⑀M, which depend on the polymer
concentra-tion ⌽P.
The experimentally observed phase diagram共see Fig. 2 in Ref. 27兲 has three characteristic features: 共1兲 The disor-dered phase neighbors the C⬜phase for all polymer concen-trations. 共2兲 Both regions of the PL phase have a limited range of polymer concentrations where they are stable. 共3兲 The thicker the film, the higher the critical polymer concen-tration where the PL appears.
We investigate the range of parameters covered by our simulations关Fig. 13共a兲兴 which give these three characteristic features. As a first reference point, the phases neighboring the disordered phase are shown in Fig. 13共d兲. The critical phase boundary C⬜/C储,1, which limits the regime where
simulations and experimental results are compatible, is shown as a bold dashed line. Figures 13共c兲 and 13共b兲 show the phases occurring for H⫽c0 and 2c0 including the char-acteristic phase boundaries C储,1/PL and C储,2/PL, respec-tively.
We look for paths in the parameter space which include all three characteristic features. This can be done by project-ing the surfaces shown in Figs. 13共b兲, 13共c兲, and 13共d兲 on each other, which is done in Fig. 13共e兲. The paths have to fulfill the following three conditions: 共1兲 They should com-pletely lay in the C⬜region and should not cross the C⬜/C储,1
boundary. 共2兲 They should first cross the C储,1/PL and 共3兲
then the C储,2/PL boundary.
The gray region displayed in Fig. 13共e兲 centers at ⑀M
⫽6.0 and corresponds to a region in the experimental phase FIG. 12. Effect of the strength of the surface field on the lateral density distribution in the wetting layer. Histograms of the A density at the surface (z⫽1) are shown for different surface interactions⑀M. Simulations have
been done in 32⫻32⫻13 large simulation boxes with H⫽12. With decreas-ing ⑀M, the A blocks are more strongly attracted to the surface and the
lateral homogeneity of the wetting layer increases.
FIG. 13. 共a兲 Range of parameters covered by our simulations. The planes
⑀AB⫽6.3 and 6.5 correspond to Figs. 8 and 9, respectively. The dark gray
surfaces are displayed in detail in 共b兲–共d兲. 共b兲 Surface reconstructions formed in films with H⫽12 as function of⑀ABand⑀M.共c兲 Same as 共b兲 for
H⫽6. 共d兲 Surface reconstructions next to the region of the disordered phase.
This region is approximatively bounded by H⫽4. Lines indicate phase boundaries.共e兲 The phase boundaries C储,2/PL, C储,1/PL, and C⬜/C储,1taken
from共b兲, 共c兲, and 共d兲, respectively. The arrows␣andcorrespond to two possible models of how the interaction parameters can change with chang-ing polymer concentration⌽P. Both models cross the gray region where a
diagram centered at⌽P⫽0.59.30Therefore, the most simple
way to parameterize such a path is given by the linear rela-tion ⑀M⫽⑀M
melt⌽
P, with⑀M
melt⫽10⫾1, which is displayed in
Fig. 13共e兲 as arrow ␣. The discrepancy with our previous publication27 is due to the fact that the experimental phase diagram was presented in units of the chloroform vapor pres-sure, whereas here we use the measured polymer concentra-tion from Ref. 30. Nevertheless, both values are close and the physical picture remains the same. By adjusting a single parameter the measured and calculated phase diagrams can be perfectly matched. In particular the predicted ⑀M values
for the onset of the PL phase at H⫽6 and 12 agree nicely with the experiments. Remarkably, the experiments can be described by a parametrization where only⑀M changes with
⌽P while ⑀AB is constant. Other possibilities would be
ar-rows like , where both parameters, ⑀AB and ⑀M, change with⌽P. The choice of the path␣is supported by the ex-perimental observation that the SBS/chloroform system stud-ied by Knoll et al.30forms cylinders in the bulk in the whole range of accessed polymer concentrations. This suggests that the influence of⌽P on⑀ABis rather weak. This is consistent
with the fact that the gray region in Fig. 13共e兲 has a consid-erable larger extent along the ⑀M axis than along the ⑀AB
axis.
B. Effect of the wetting layer
In Fig. 14 depth profiles of the laterally averaged A den-sity are compared. The profiles of the film forming the lamella and wetting layer surface reconstruction coincide with that of the film forming parallel cylinders when the profile corresponding to the lamella is shifted by c0 and that of the wetting layer is shifted by c0/2. This indicates that the wetting layer can be regarded as a half lamella with thickness
c0/2. Furthermore, both the lamella and the wetting layer
screen the surface field and the depth profile below them is that of a film forming cylinders oriented parallel to the inter-face. Effectively, the A-wetting layer induces a B-rich layer at c0/2, which corresponds to a situation at the interface of a
film which preferentially attracts the B component. The lamella screens a strong surface field in a similar way.
In experiments with supported films the interactions at the air/film and the film/substrate interface are in general different. In a situation where one interface attracts the A and the other the B component, the formation of a wetting layer at one interface can lead to a situation where the film can be treated as having effectively both interfaces attracting the B component. Therefore, the phase diagram measured by Knoll
et al.30 can be well described in simulations with equal in-terfaces, although the experiments clearly indicate the pres-ence of an A-wetting layer at the film/substrate interface and the preferential attraction of the B component at the air/film interface.
C. Comparison with cylinder forming diblock copolymers
The influence of the molecular architecture on the ob-servered phenomena can be studied by comparing our results on A3B12A3 triblock copolymers with the behavior of the
corresponding A3B6 diblock copolymer studied by Huinink et al.21 The comparision is made easy since in both studies the same parameters were used and we varied 共in addition兲 the interaction parameter⑀ABonly slightly. For both systems
we are well in the part of the phase diagram where cylinders form in the bulk.
At first glance, no utterly significant difference between the phase diagrams of the A3B6diblock copolymer共Fig. 5 in
Ref. 21 and Fig. 4 in Ref. 22兲 and our A3B12A3 triblock copolymers is visible. Only the position of phase boundaries between different phases differs slightly. This fact leads us to the important conclusion that the observed phenomena and mechanisms are present in many cylinder-forming block co-polymers. In particular, the molecular architecture plays only a minor role and enters only via the specific values of the interaction parameters. This is further corroborated by results of Wang et al.24 obtained with Monte Carlo simulations, which also show a similar phase behavior for cylinder-forming systems in thin films.
D. Comparison with lamella-forming diblock copolymers
We note that the orientation behavior of the cylinders is analogous to the phase behavior of lamella-forming diblock copolymers as both arc controlled by the interplay between the surface field and confinement effects.4,5 Thus, the se-quence C储→C⬜→C储 at steps between terraces corresponds to the sequence L储→L⬜→L储.38 Second, in cases where the two confining surfaces favor different orientations (L储,L⬜) the two orientations can coexist and a hybrid 共or mixed兲 structure 共HY兲 forms39 which is similar to cylinders with necks.16We note that in such a HY structure the bulk micro-domain structure is preserved and a grain boundary is stabi-lized in the thin film by the antisymmetric surface field. Fur-thermore, a disordered phase has been reported for ultrathin films of lamella-forming diblock copolymers3,10 and is in nice agreement with our findings and the experiments of Knoll et al.27,30In addition to the alignment effect,
hexago-FIG. 14. Depth profiles of the laterally averaged A density具A典x,yin thin
films with H⫽54 and⑀AB⫽6.5 for different surface fields. The depth
pro-files are shifted according to z⬘⫽z for C储(⑀M⫽6), z⬘⫽z⫺3 for W (⑀M
⫽⫺4), and z⬘⫽z⫺6 for L (⑀M⫽30). The solid line is a spline through the
L data.
nally ordered cylinders can adopt to the planar surface by formation of surface reconstructions 共W, PL, L兲 which also dominate the phase behavior in thin films.
V. CONCLUSIONS
Though based on a rather simple microscopic model, our DDFT simulations correctly predict a phase diagram with intriguing complexity. This close match with the experimen-tal data together with the large range of parameters covered by both experiments and simulations, make us believe that we have identified the relevant physical parameters and the mechanisms governing structure formation in the films cyl-inder forming block copolymers. In particular, the large pa-rameter space covered allows us to distinguish the effects of the two constraints being simultaneously present in a thin film situation: the surface field and the film thickness. Our results also reveal the mechanism of how both interplay.
We identified the deviations from the bulk structure, both in the vicinity of surfaces and in thin films of cylinder-forming block copolymers as surface reconstructions. To-gether with what is known for lamella-forming systems our results give evidence for a general mechanism governing the phase behavior in thin films of modulated phases: The strength of the surface field and the deformability of the bulk structure determines how the system rearranges in the vicin-ity of the surface. This causes either an orientation of the bulk structure or the formation of surface reconstructions. The stability regions of the different phases are modulated by the film thickness via interference and confinement effects.
This concept along with the presented method might provide the means to understand and eventually control a wealth of thin film structures in a wide class of ordered flu-ids, such as linear and star multiblock copolymers as well as surfactant based fluids.
ACKNOWLEDGMENTS
We thank G. Krausch, A. Knoll, and M. Hund for dis-cussions and help. We acknowledge support from the Deut-sche Forschungsgemeinschaft 共SFB 481兲, the Volkwagen-Stiftung, and the NWO-DFG bilateral program.
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